Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 012, 33 pages      arXiv:1705.09469      https://doi.org/10.3842/SIGMA.2018.012

$k$-Dirac Complexes

Tomáš Salač
Mathematical Institute, Charles University, Sokolovská 49/83, Prague, Czech Republic

Received June 01, 2017, in final form February 06, 2018; Published online February 16, 2018

Abstract
This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.

Key words: Penrose transform; complexes of invariant differential operators; relative BGG complexes; formal exactness; weighted jets.

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